Lecture 2: The intertemporal approach to the current account cont'd.
Open economy macroeconomics, Fall 2006
Ida Wolden Bache
August 28, 2006
i
What determines the world interest rate?
•  A two-region world economy
—  Abandon assumption of small open economy
—  World is composed of two countries:  Home and Foreign (*)
—   Ignore government spending
•   Endowment economy
—  Global market clearing requires
Yt + Yt* = Ct + Ct or
St + st = o
2
-  Walras's law: with two markets, only need to check that one of them clears to verify general equilibrium
—  World market equlibrium in two-period model
Sl(r) + St(r) = 0
3
How does saving depend on the interest rate?
— Key concept: the intertemporal elasticity of substitution
u\C)
a(C) =
Cu"(C)
—  The sensitivity of intertemporal consumption allocations to interest rate changes depends on a
—  Class of preferences with constant elasticity of intertemporal substitution (isoelastic preferences)
i    1 C1-^ - 1 <C) =
a
—  Special case: a = 1 —> u{C) = In C
4
—   Euler equation with isoelastic preferences
C~° = ß(l + r)C2"-C2 = ßa(l + r)CTCi
Ci
—  Define subjective rate of time-preference (5
1             ,     1-/3 /3 =-----------=*►<* =----------
l + <5                        /3
—   Euler equation becomes
a
C2= n + r
Consumption grows over time if the real interest rate exceeds the subjective rate of time-preference i.e., if r > ö
5
Derivation of the saving schedule (saving as a function of interest rates) with isoelastic preferences
- Substitute in for budget constraint Q? = (1 + r)(Y± — C\) + I2 'n the Euler equation
Gl   =   ß~a{l + r)-aC2
=   /?"-(!+ r)--((l + r)(Y1-C1)+Y2)
Differentiate
=   -<x(l + r)-^-1/?-^ ((1 + r)(n - CO + Y2)
dCl                     /-     ,       x-cr-lo-CT
+/3"CT(1 + r)-%Yl - d)
-ß-^l + r)1-^
ar
Collect terms and substitute in for Q?
dCi _ -cr(l + r)-(J-1ß-(7C2 + ß~a(l + r)-a(Y1 - d) dr   ~                         1 + ß~(7(l-\-r)1-(7
Divide through by ß~a(l + r)-a
dcľ = -^ + (n - Cl)
dr   "~ /3er (1 + r)a + 1 + r Substitute in for C2/C2 from Euler equation
dd = -*& + (n - co
dr             ^ + 1 + r
7
— A rise in r has ambiguous effect on C\ and hence on saving S\ = Y\ — C\
*   Substitution effect: an increase in the interest rate makes future consumption relatively cheaper —> lower current consumption (increase saving)
*   Terms-of-trade effect:
•   if Y\ — C\ > 0 interest rate increase makes the country richer (improves intertemporal terms of trade) —> higher current consumption (lower saving)
•   if Y\ — C\ < 0 interest rate increase makes the country poorer (worsens intertemporal terms of trade) —> lower current consumption (higher saving)
8
— Closed form solution for consumption with isoelastic preferences
ß^l + r^d   =   (l + r)(Y1-Ci) + Y2
(ßa(l + r)a + l + r)C1   =    (l + r)Y1 + Y2
Y2
ci = „„,. , w-1 , . n +
)Sť7(l + r)ť7-1 + l V         1 + r.
*   Three effects of an increase in the interest rate
1.  Substitution effect (-=-) 2.  Income effect (+) 3. Wealth effect (-=-)
*   Income effect dominates substitution effect if a < 1
*  With a = 1 (log case) income and substitution effects cancel —> dC\/dr always negative
*   Income effect + wealth effect = terms of trade effect
Interest rate, r S*
Home saving, S
Foreign saving, ST
Figure L5
Global exchange equilibrium
Comparative statics
-   Increase in Y\ shifts saving schedule SS1 out —> r falls
-   Increase in Y<i shifts saving schedule SS' in —> r increases
-   Increase in /3 shifts saving schedule SS7 out —> r falls
Immiserizing growth' (Bhagwati, The Review of Economic Studies Vol 25, 1958): Under certain circumstances, economic expansion may harm the growing country. The increase in output might lead to a sufficient deterioration in the terms of trade to reduce the real income of the growing country
11
• World equilibrium in the two country model with investment
-  Allow for different productivity levels
Y = AF(K),       F* = A*F*(K*)
—   Equilibrium requires
n + n* = Ci + cí + h + it
or
Si + SJ = ii + ij or
CAX + CAJ = 0
12
7585
Investment schedules
A2F\K1 + li) A$F*'{K{ + I{)
r r
Production functions concave: investment schedules slope downward
Saving schedules with isoelastic preferences
— Substitute in for budget constraint in the Euler equation
Ci   =   ß-a(l + r)-aC2
=   ß-a(l + r)
—a
A2F(K1 + h) + K1 + I1 +(1 + r) (AiF(ifi) -d- h))
13
Differentiate
=   -<r(l + r)-a-Lß-aC2
dCl                     f,      .        X-CT-lo-CT.
ÓjT
+/3-CT(l + r)"- {AXF{K{) -d- h))
o,-,    ,      \l-adCl
-ß-a(l + r)
+ß~a{l + r)-a(A2F'(K2) + 1 - (1 + r))
dr
dlľ
L dr =0 (envelope theorem!)
Divide through by /3_cr(l + r)~a and substitute in for C2/C2 from Euler equation
dd = -*£& + (n - ci - /p
dr                 ^ + 1 + r
Note!   For a given value of the current account balance, the slope of the saving schedule is the same as in the endowment economy
14
Interest rate, r                                                                      lnterest rate-r
Home saving, S                                                             Foreign saving, S*
Home investment, I                                                       Forei9n investment,
Figure 1.7
Global intertemporal equilibrium with investment
Comparative statics — Fall in ß shifts saving schedule SS' inwards —> r increases
— A rise in future home productivity A2 * Shift in investment schedule II'
A2Ff{Kľ + Ii) = r dA2Ff(K2) + A2F"{K2)dI1 = 0
dlľ dA2
F'{K2)
r constant
A2F"(K2)
>0
* Shift in saving schedule SS'
)Sť7(l + r)ť7-1 + l
dCx dA2
AlF(Ki) - h +
F(K2)
A2F(K1 + /Q - I2 1 + r
r constant
/3(J(l + r)(J + 1 + r
>0
16
Interest ratet r
Interest rate, r
Home saving, S Home investment, I
Foreign saving» S* Foreign investment,
Figure 1.8
A rise in future Home productivity
10.0
E.O
5 0
4 0
2.0
0 0
-2.0
Figure 1. United States' 10-Year Bond Rate (Percent a year)
U
~~ Expected In3. — Actual Inf.
-P-V
K
Ajw
■v

W^
,.«       i
J-4
\
-4.0   I iiiiiiii im iiiiiiiiiiiiii 'ii. in   ill ill   iii iii   in   ii   in   ii   in   ii   m   nim   ii.in
1957 1959 1961 1963 1965 1967 1969 1971 1973 1975 1977 1979 1981 19B3 19E5 19S7 19S9 1991 1993 1995 1997 1999 2001 2003 2005
•   Case study: The U.S. real interest rate and current account imbalances
•   The Global Saving Glut and the U.S. Current Account Deficit (Speech by Chairman of the Board of Governors Ben Bernanke, April 14 2005. Can be downloaded from: http://www.federalreserve.gov/boarddocs/speeches/2005/20050414/default.htm)
—  A significant increase in the global supply of saving helps explain the increase U.S. current account deficit and the low level of world real interest rates
-  What has caused the increase in global saving?
*   Strong saving motive of rich countries with ageing populations
*   Low prospetive returns to domestic investment in many mature industrialised countries (due to e.g., slowly growing workforces and high capital-to-labour ratios)
19
*  Movement from large deficit to large surplus in the current account positions of developing countries
•   Response to the financial crises in the 1990s
•   High oil prices
-   High inflow of capital to the U.S. because of technology boom and high productivity in 1990s
—   Low U.S. saving because of high stock prices, increase in housing wealth and later, low real interest rates.
20